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Contact structures and reducible surgeries

Part of: Low-dimensional topology Differential topology

Published online by Cambridge University Press:  24 September 2015

Tye Lidman  and
Steven Sivek
Tye Lidman
Affiliation:
Mathematics Department, The University of Texas at Austin, RLM 8.100, 2515 Speedway Stop C1200, Austin, TX 78712-1202, USA email [email protected]
Steven Sivek
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA email [email protected]
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Abstract

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus $g$ must have slope $2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

Keywords

cabling conjectureDehn surgerycontact structuresStein fillings

MSC classification

Primary: 57M25: Knots and links in $S^3$ 57R17: Symplectic and contact topology
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 1 , January 2016 , pp. 152 - 186
Copyright
© The Authors 2015 

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